Unlinking particle system datablocks from python
Copy the particle system settings

Here is a simple test script on what I believe you want to achieve.  Select object with at least one particle system.  Copies the context object, copies the settings of particle system 0, changes the start frame on copy to 22 and sets this to particle system 0 on copy.

import bpy

context = bpy.context
scene = context.scene
ob = context.object
# copy object 
obcopy = ob.copy()
# copy particle settings
ps = ob.particle_systems[0].settings.copy()
ps.frame_start = 22
# new settings on copy
obcopy.particle_systems[0].settings = ps
# link to scene
scene.objects.link(obcopy)